Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1
![Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1 Class 11 Maths NCERT Solutions for Chapter 13 Limits and Derivatives Exercise 13.1](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhr6QYTi-AhHRqq0Z7XnjDeLQfoOu5qTHcKNlKv0wgHYk325_AuT8uwj9PQk2OOn3GK0rmNNXQUYjz66aRMCpQPF-gDV6dOXUeBaWaFYFmJDaB4a359XnflPcjQZyXTfpvxr6gp6vEeqiBz8vKXrhYpFHhc_32nJUXFZ74CE2WO23jh-BCeNFv1fsFo/w663-h313-rw/ncert-solutions-for-class11-maths-limit-and-derivatives-exercise-13-1.jpg)
Limits and Derivatives Exercise 13.1 Solutions
1. Evaluate the Given limit (x + 3)
Solution
(x + 3) = 3 + 3 = 6
2. Evaluate the Given limit : (x - 22/7)
Solution
3. Evaluate the Given limit :
Ï€r2.
Solution
4. Evaluate the Given limit :
(4x + 3)/(x - 2)
Solution
5. Evaluate the Given limit :
(x10 + x5 + 1)/(x – 1)
Solution
6. Evaluate the Given limit :
[(x + 1)5 - 1]/x
Solution
7. Evaluate the Given limit :
(3x2 - x - 10)/(x2 - 4)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .
8. Evaluate the Given limit
(x4 -81)/(2x2 - 5x - 3)
Solution
At x = 2, the value of the given rational function takes the form 0/0 .
9. Evaluate the Given limit :
(ax + b)/(cx + 1)
Solution
10. Evaluate the Given limit : ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7Mt7G_lZtcr36y0X5h_DW2T_8lU071P6SgqtpI-X7iqgCI1CKsLIROqk8zSoOuAk7WVVP_60wuGk7hSlKla5Kg9CASYoIp5rWHq0CHZP0ZuqJTemMQ5z2OqmpGoACDRtG3cjCdb2lacA3VuAuvzc-YJ1LraRVwrc3gT4lzSKhYgMkFXe52cCE7LNp/w94-h55-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2017.JPG)
Solution
At z = 1, the value of the given function takes the form 0/0 .
Put z1/6 = x so that z →1 as x →1.
11. Evaluate the Given limit
(ax2 + bx + c)/(cx2 + bx + a) , a + b + c ≠ 0 .
Solution
= 1 [a + b + c ≠ 0 ]
12. Evaluate the Given limit : ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJa1a45g8ynWyFvc5Hp6CDPjK1S3k3UI2bHnTumjwSVBEfycwoeISXFJx3EwVQLsjrI_-brM6Ih_omC5GaEEDr59NLzYXIjaRZyd2xzwadeV0xo5ddDDhHCfQW-mH3gZoGCY840CNZp6JCcnvSr-7IG_XDhKiXWGWwNCRGSWLkc1IamP9iDpRygdZP/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2021.JPG)
Solution
At x = -2, the value of the given function takes the form 0/0 .
13. Evaluate the Given limit
sin ax/bx
Solution
At x = 0, the value of the given function takes the form 0/0 .
14. Evaluate the Given limit :
sin ax/sin bx, a,b ≠ 0
Solution
At x = 0, the value of the given function takes the form 0/0.
15. Evaluate the Given limit
[sin(Ï€ -x)]/Ï€(Ï€ - x)
Solution
16. Evaluate the given limit
cos x/(Ï€ - x)
Solution
17. Evaluate the Given limit :
(cos 2x - 1)/(cos x- 1)
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now,
18. Evaluate the Given limit
(ax + x cos x)/b sin x
Solution
At x = 0, the value of the given function takes the form 0/0.
Now,
19. Evaluate the Given limit
sec x
Solution
20. Evaluate the Given limit
(sin ax + bx)/(ax + sin bx) a,b, a+b ≠ 0.
Solution
At x = 0, the value of the given function takes the form 0/0 .
Now,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0KrJ6gYzEbTlN2WY37n81JrmKCH4SWwsNA14BGoEfN_lNW1lTI7QC2-CU-YJYHE_FdSX0KCrEJ0Gi-iWGA3L_APIYtzd04CuZQCi7HEnw66ST09AL6__ZJ0ZXX8Jxjg6Qec0is_WMgG0D0VspBLoSm3PtaILMVHwo9DF3avDX2p6rj8JhEKg_NSLM/w502-h380-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2030.JPG)
Now,
21. Evaluate the Given limit
(cosec x - cot x)
Solution
At x = 0, the value of the given function takes the form ∞→∞.
Now,
Now,
22. Evaluate the Given limit
Solution
At x = π/2 , the value of the given function takes the form 0/0.
Now, put x - Ï€/2 = y so that x → Ï€/2, y → 0.
Solution
The given function is ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHeT04sXYcXa6YxFsJwICoBMm48srf0F2Txu-kuycsj6tYfhA8dlN3o0AGBzc7We-Z0IePYVSn12mmOU8LLwT8pO8-wEEtgvY78O6mLq6nAp-GineRhxUJg61z5Je7ZNaihMOucH_XtHoz5G6sP6IaucyXZtrD_8NuVQ1IPFhmQOz81KmGW3G84sNK/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2034.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaDwBJND4CzFj6KUXfTElgF8KsP81OaCqVtwnukba3qeF2YQ6IQrpTyfIFSLgF548uMMLm2zB4w5jp9jPyW3sQKiDasclc27dynJTzrKvAS_FrQO5Vkh3covpKBTVuMWupXuzFbx3rpMfTs8qNmgIxbWGxb-IoVvPqb4u52AIaqRMC6rmvi2KfJeRt/w257-h267-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2035.JPG)
24. Find
f(x) where ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQ6_Iprw6psxIhGbmA2rydpnTtXvVKnYBQxlFCJW4DoBQCkWzZzeRP6BrTtcc9OhYLP5U7RNoo2UTl46SBzk-srKNNi0CapMzRKimeDOCoSVfIGkt_LPD9DOb9orsBk27mvCIyo2_3LZWizKpj6V9zVjrEaz1ymunRXrN1l_1_tUj6qB-bTS1-Cf0_/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2036.JPG)
Solution
The given function is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUDfwXaBs9hASLXqWDhDN4wSjDOxDNaSRu1JlOS-4VQUjKu7S8HCsFJ0PkVG6YqcdlP3Bm2I_rx3Spw0C90ECgbkegSrhCXdNglT10Veq4dEx-rCD6gvzyTn1LFknXOPRMWHXzjM0HkZIyZgy7N_F7NfTQCW8hoH-IcS8DBB_7bCW_RiNIy1iTpH5V/w309-h206-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2037.JPG)
25. Evaluate
f(x) where ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyYjfhTyoI6DOQ7DHbLckgq_8la4i2TKdLsNbMCyc5Lf9zgBhgLTwXyE7EvZmXl_UlTsmojxSLoHok95l_9wMQnxJWjALLDjQBA15dHlQTP5Yj00w4lyP2oMwOyrX6l3y2v6lkOQLArw9merFzM17r0hyqD48LzR-wIs3z2e3nJoe8_kNIFQ_WL8Ni/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2038.JPG)
Solution
It is observed that
Hence,
26. Find
f(x) where ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCnXD8nsTRfWYiG_mV6kkojjjExmfJIPgjWzh5yQkH7qrLuEmrMYBqrKyN6C7SGUsDwyW844eAhBm9xd5ky0hsH0r9MpU941z-wtU_awFL_O5MMz9ygU-rZiO_nKGsaufEjRx3pKX2pc8l5vACbkFoLCsP0Kx7LTkPjMm3bUIrPCPbOCEVCcHyriv7/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2040.JPG)
Solution
The given function is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXskLfaAh4MaaoiLNmiSRvtinx7Yv5IV3vzTfwVFqiOmFuVvZY9dXcLm-IU_W7FXdQSGeiK4tD9YoBeXo0-bgIZ5b7pORVKXvynFZCFfuV6iDHloYVzwhDY731v94_hdrbwrah8OoxXnJ9pXM9NAVDLHgXKRcnJlHLbTPlsoeY5TapvQJf2Xen9anN/w287-h389-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2041.JPG)
It is observed that
f(x) ≠
f(x).
Hence,
f(x) does not exist .
It is observed that
Hence,
27. Find
f(x) , where f(x) = |x| - 5
Solution
The given functions is f(x) = |x| - 5.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3FtxLxyS10or59_Af2GaD_SKvwToHP3lEANRakZCDUlWVuast7TAnIP9MSFX-DtfgFZ157LE5tvmUVp9xW5OFrNXEs1lLHZ3W_M212NzaibY1GiupWTQBYaoQiFEovQC--cWvD1m-Mah27oQpSLJXqqW_jYzXtTVyo1xMlp0GcWVoq3UZxadfBm8V/w284-h294-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2043.JPG)
28. Suppose
and if
f(x) = f(1) what are possible values of a and b?
Solution
The given function is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH0w4xFvSMIK33x-sqhk_o-vYtm-DzMlpjxoAhmDmx1jcOfxUJ3jGIAHRMod1MQ7ui0XD-SHjN8m-8uqX2mixp97Rma3CP3gjbpSQyRJ4-jC5bAAyMI_orNZB0VtX3EnWCXlEiT0BHNoRa2uKhzlmm9HSm2Jtv47_czZLDbi4PVV12Oq47_cX9XFi-/w192-h88-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2044.JPG)
f(x) =
(a + bx) = a + b
f(x) =
(b - ax) = b - a
f(1) = 4
It is given that
f(x) = f(1).
∴
f(x) =
f(x) =
f(x) = f(1)
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.
f(1) = 4
It is given that
∴
⇒ a + b = 4 and b - a = 4
On solving these two equations, we obtain a = 0 and b = 4.
Thus, the respective possible values of a and b are 0 and 4.
29. Let a1, a2, . . ., an be fixed real numbers and define a function f ( x) = (x − a1) (x − a2)...(x − an). What is
f(x) ? For some a ≠ a1, a2, ..., an, compute ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh00M7zsxsJ1GVlNsK3F4hYhoLqKSiF1m08-KQ1UatihdNbzR5XBo-ngSvaRu-iea02d2UtZCg29A-kWzhiHpgh7qFkT-XVmk-xWcmS4azYxy2WnGgS5lm559SlQ8q380uxx2EzqvlDIJc6Y2ntTbA0cbOGz21XsctTw34y2DPAVz_D_g6A0paKn9hr/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2046.JPG)
Solution
The given function is f(x) = (x – a1)(x – a2) … (x - an).
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcXJ6voCS-z8IqxYY_pduvuCa0vEuRDdLhFUzcENuxh9ABaUwIq0w-m4dWakkAEiyaLWbaUM3FT_-oC3rGJaHLILYe51lGRyzCDoI9EJTXDSMPqNrWOXrF48Sc8PXYtQ6bRPYtKHj0NPMRYwfhN9jCDImUScWjzseuwEQif24jERhmEi-JOOg32Uq2/w353-h289-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2047.JPG)
30. If
For what value (s) of a does
f(x) exists ?
Solution
The given function is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiZ80FPNUbSGhwTNfTFc_Vj1ctf4Zs9KKhMmn3nQSEw86bOBm5PkA8NZvtFvU5BQR5xEPO5c7474LPw69fYv9XqYcvPqQJCtaTMzAugrcYBKSsIaxj5H9sn0SxAItuIkteGqyPuFAea735WiBFxNBoEqf0ahvB-iD5skwzlJq6pU-pUVVyxwG2yis7/s1600-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2048.JPG)
When a = 0,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_AiPmenYTCRZ8ofU7NqekawNXJ_IXqpSlsRdFSgJhUEbCnR9HoOWHV5VRx2cXh1gWSXDLTYm9vXHIpR1DIe0unEi5souotoDILnnMoA3SAnguCOISyG5zr_aGqaDb9-dD_NeCMwcddLq2Meix9Gp5JUF8_MKD9SK_3PUbu5tYk-jn0PfyW1gN44rh/w360-h960-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2049.JPG)
When a = 0,
31. If the function f(x) satisfies
[f(x) - 2]/[x2 -1] = π , evaluate
f(x).
Solution
32. If ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMgD6wmqg7FibG2JdDNP6H862MPlJjxQ6UpP0SLPugy0LSPQeZMKcCA9tRY3cl0-1nmEm5qfVDKLOUFXXWU1UCgsxpsobNBaFeLN1FnN8NmRwZWhlDUL-VgUcboK7VwnrHIhawZoH_yGQS4tad3uoqv0fhECCttBFzEtTmeSaW4Q2Q6W0BOETDl_cW/w249-h78-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2052.JPG)
For what integers m and n does
f(x) and
f(x) exist ?
For what integers m and n does
Solution
The given function is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaoKs6My7vtZq6KLWSrnBxjRA-h-MpgEHYAluRZWtLBTJybIDbOc0xWcMVaW0oZ-oosoS5sQFsLE6wX9pgLwlw21b4h229xynsYD5zUcyIHkfRnB_Nyw77MU_dM8I38I4AJFoC7MYoUEik00ALVmrnjDAMe_q6AXSy3QDCPmJDd-3nia-RaAQY9xdk/w367-h532-rw/NCERT%20Solution%20For%20Class%2011%20Maths%20Chapter%2013%20Limits%20and%20Derivatives%20%20Exercise%2013.1%20img%2053.JPG)